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We prove that $\\Fl^a_\\la$ have rational singularities, are normal and locally complete intersections, and construct a desingularization $R_\\la$ of $\\Fl^a_\\la$. The varieties $R_\\la$ can be viewed as towers of successive $\\bP^1$-fibrations, thus providing an analogue of the classical Bott-Samelson-Demazure-Hansen desingularization. We prove that the varieties $R_\\la$ are Frobenius split. 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